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Exact real trinomial solutions to the inner and outer Hele-Shaw problems. (English) Zbl 1317.76019

Summary: We find an explicit representation of the evolution of \(t \mapsto\gamma_t=\{z(\zeta,t),\zeta\in\mathbb C,|\zeta|=1\}\) of the contour \( \gamma_t=\partial\omega_t\) of fluid spots \(\omega_t=\{z(\zeta,t),|\zeta|<1\}\) for \(t>0\) or \(t<0\) in the Hele-Shaw problem with a sink \((t>0)\) or a source \((t<0)\) localized at point \(z(0,t)\) described by trinomials \[ z(\zeta,t)=a_1(t)\zeta+a_N(t) \zeta^N+a_M(t)\zeta^M,\text{ where }M=2N-1,\text{ and integer }N \geq 2, \] for the classical formulation of the problem when \(\omega_t\) is within \(\gamma_t\) (inner Hele-Shaw problem), or by \[ z(\zeta,t)=a_{-1}(t)\zeta^{-1}+a_N(t) \zeta^N+a_M(t)\zeta^M,\text{ where }M=2N+1,\text{ and integer }N \geq 1, \] for the outer Hele-Shaw problem when \(\omega_t\) is outside of \(\gamma_t\). We obtained a sufficient condition for univalence of real trinomials, improving a result found by St. Ruscheweyh and K. J. Wirths [Ann. Pol. Math. 28, 341–355 (1973; Zbl 0244.30013)]. A sufficient condition is also found for functions used in the outer problem.

MSC:

76B07 Free-surface potential flows for incompressible inviscid fluids
30C50 Coefficient problems for univalent and multivalent functions of one complex variable

Citations:

Zbl 0244.30013
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References:

[1] Ben Amar M.: Exact self-similar shapes in viscows fingering. Phys. Rev. A 43(10), 5724-5727 (1991) · doi:10.1103/PhysRevA.43.5724
[2] Brannan D.A.: Coefficient regions for univalent polynomials of small degree. Mathematika 14(2), 165-169 (1967) · Zbl 0193.37004 · doi:10.1112/S0025579300003764
[3] Cohn A.: Uber die Anzahl der Wurzeln einer algebraischen Gleichung in einem Kreise. Math. Zeitsch. 14(1), 110-148 (1922) · JFM 48.0083.01 · doi:10.1007/BF01215894
[4] Cowling V.F., Royster W.C.: Domains of variability for univalent polynomials. Proc. Am. Math. Soc. 19(4), 767-772 (1968) · Zbl 0159.10501 · doi:10.1090/S0002-9939-1968-0227392-2
[5] Dieudonné J.: Recherche sur quelques problèmes relatifs aux polynômes et aux fonctions bornées d’une variable complexe. Ann. Sci. l’ENS 48, 247-358 (1931) · JFM 57.0355.02
[6] Galin, L.A.: Unsteady filtration with a free surface. Dokl. Akad. Nauk USSR 47, 246-249 (1945) (in Russian) · Zbl 0061.46202
[7] Gustafsson B.: On a differential equation arising in a Hele-Shaw flow moving boundary problem. Arkiv Mat. 22(1-2), 251-268 (1984) · Zbl 0551.30037 · doi:10.1007/BF02384382
[8] Hele-Shaw, H.S.: Flow of water. Nature 58(1489), 34-36 (1898) (520) · Zbl 0551.30037
[9] Howison S.D.: Cusp development in Hele-Shaw flow with a free surface. SIAM J. Appl. Math. 46(1), 20-26 (1986) · Zbl 0592.76042 · doi:10.1137/0146003
[10] Howison S.D., King J.: Explicit solutions to six free-boundary problems in fluid flow and diffusion. IMA J. Appl. Math. 42, 155-175 (1989) · Zbl 0673.76099 · doi:10.1093/imamat/42.2.155
[11] Huntingford C.: An exact solution to the one-phase zero-surface-tension Hele-Shaw free-boundary problem. Comput. Math. Appl. 29(10), 45-50 (1995) · Zbl 0827.76017 · doi:10.1016/0898-1221(95)00044-Y
[12] Kuznetsova, O.: On polynomial solutions of the Hele-Shaw problem. Sibirsk. Mat. Zh. 42(5), 1084-1093 (2001) (English translation: Siberian Math J 42(5), 907-915, 2001) · Zbl 0998.34071
[13] Lamb H.: Hydrodynamics, 3rd edn. Cambridge University Press, Cambridge (1906) · JFM 36.0817.07
[14] Leibenson L.S.: Oil Producing Mechanics, Part II. Neftizdat, Moscow (1934) · JFM 60.1387.03
[15] Nishida T.: A note on a theorem of Nirenberg. J. Differ. Geom. 12(4), 629-633 (1977) · Zbl 0368.35007
[16] Polubarinova-Kochina, P. Ya.: Concerning unsteady motions in the theory of filtration. Prikl. Mat. Mech. 9(1), 79-90 (1945) (in Russian) · JFM 48.0083.01
[17] Rahman Q.I., Szynal J.: On some classes of univalent polynomials. Can. J. Math. 30(2), 332-349 (1978) · Zbl 0381.30005 · doi:10.4153/CJM-1978-030-2
[18] Ruscheweyh S., Wirths K.J.: Uber die Koeffizienten spezieller schlichter polynome. Ann. Pol. Math. 28, 341-355 (1973) · Zbl 0244.30013
[19] Saffman P.G., Taylor G.I.: The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid. Proc. R. Soc. Lond. Ser. A 245 281, 312-329 (1958) · Zbl 0086.41603 · doi:10.1098/rspa.1958.0085
[20] Stokes, G.G.: Mathematical proof of the identily of the stream-lines obtained by means of viscous film with those of a perfect fluid moving in two dimensions. Br. Assoc. Rep. 143 (papers, V, 278) (1898) · JFM 29.0645.04
[21] Vinogradov, Yu.P., Kufarev, P.P.: On a problem of filtration. Akad. Nauk SSSR Prikl. Mat. Mech. 12, 181-198 (1948) (in Russian) · Zbl 0032.27901
[22] Vinogradov, Yu.P., Kufarev, P.P.: On some particular solutions of the problem of filtration. Doklady Akad. Nauk SSSR (N.S.) 57, 335-338 (1947) (in Russian) · Zbl 0029.03601
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